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Mirrors > Home > MPE Home > Th. List > Mathboxes > ispointN | Structured version Visualization version GIF version |
Description: The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ispoint.a | ⊢ 𝐴 = (Atoms‘𝐾) |
ispoint.p | ⊢ 𝑃 = (Points‘𝐾) |
Ref | Expression |
---|---|
ispointN | ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ispoint.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | ispoint.p | . . . 4 ⊢ 𝑃 = (Points‘𝐾) | |
3 | 1, 2 | pointsetN 35027 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑃 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}}) |
4 | 3 | eleq2d 2687 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}})) |
5 | snex 4908 | . . . . 5 ⊢ {𝑎} ∈ V | |
6 | eleq1 2689 | . . . . 5 ⊢ (𝑋 = {𝑎} → (𝑋 ∈ V ↔ {𝑎} ∈ V)) | |
7 | 5, 6 | mpbiri 248 | . . . 4 ⊢ (𝑋 = {𝑎} → 𝑋 ∈ V) |
8 | 7 | rexlimivw 3029 | . . 3 ⊢ (∃𝑎 ∈ 𝐴 𝑋 = {𝑎} → 𝑋 ∈ V) |
9 | eqeq1 2626 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑎} ↔ 𝑋 = {𝑎})) | |
10 | 9 | rexbidv 3052 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑎 ∈ 𝐴 𝑥 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})) |
11 | 8, 10 | elab3 3358 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = {𝑎}} ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎}) |
12 | 4, 11 | syl6bb 276 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 Vcvv 3200 {csn 4177 ‘cfv 5888 Atomscatm 34550 PointscpointsN 34781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-pointsN 34788 |
This theorem is referenced by: atpointN 35029 pointpsubN 35037 |
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